Write then XX for the space XX regarded as a sheaf or trivial covering space over itself, i.e. the terminal objectXX in sheaves and hence in stacks over XX. Then by definition of stackification morphisms

X→const¯C
X \to \bar const_C

are represented by

an open cover {Ui}\{U_i\} of XX;

over each UiU_i a choice Fi∈CF_i \in C of object in CC, hence a finite set in CC;

Specifically for g:*→ΔBAut(F)≃BΔAut(F)↪Δcore(set)g : * \to \Delta \mathbf{B} Aut(F) \simeq \mathbf{B} \Delta Aut(F) \hookrightarrow \Delta core(set) the classifying morphism of a locally constant sheaf and for U→*U \to * an epimorphism on which it trivializes, we have a pasting diagram of pullbacks